I guess I need help? I started playing again now and noticed, that this Bunny NPC is selling something that cost 0N. When I "buy" this the girl is pressing some buttons on a machine and I get some sort of octopus arm. What is that used for?

It's just a quest item from the bunny girl, nothing entirely special, just talk to her again and complete the quest and the item will disappear from your inventory. It's basically just a small tutorial on how to use her gacha system.

Warning to new players:
CLOSERS is a game that adjusts several important details in almost every single patch.
So something that may have been true a month ago may not be true now. Check the date before you reply!